Cartier Operator Research Articles

On a problem of Kodama concerning the Hasse-Witt matrix and the distribution of residues

On a problem of Kodama concerning the Hasse-Witt matrix and the distribution of residues

Frobenius Splitting and Cohomology Vanishing for Schubert Varieties

Let k be an algebraically closed field of characteristic p > 0 and X be a projective variety over k. We then have the absolute Frobenius F: X -> X and an injection Ox -* F *6x given by f fP, f e O9x. If this makes Ox a direct summand in F* Ox (as an (x-module) we call X a Frobenius split variety. For such a variety the vanishing theorem for ample line bundles follows trivially from Serre's vanishing theorem. For, tensoring Ox -F*Ox by an ample line bundle L and noting that L ? F * (x = F * F *L = F * LP (projection formula) we get that the map in cohomology H'(X, L) -+ H'(X, F*LP) = H'(X, LP) is an injection. Iterating this we see that H'(X, L) injects into H'(X, LP) for every P. But, for large v the latter is zero! Thus Frobenius split varieties have quite pleasant properties. It also turns out that using duality for the Frobenius morphism, or equivalently, the Cartier operator, one can give a manageable criterion for Frobenius splitting. The point here is the local nature of duality. The relevance of the compatibility of local and global duality was suggested to us by Grothendieck's proof of H'(X, Y) = 0 for a noncomplete variety X of dimension n ([4], Theorem 6.9) and by Kempf's paper [10]. By this criterion and the Bott-Samelson-Demazure desingularisation of Schubert varieties, it follows very easily that Schubert varieties in characteristic p are Frobenius split. The following vanishing theorem is then an immediate consequence. Let G be a reductive group over the field k (of arbitrary characteristic, zero or positive). Let Q be a parabolic subgroup and X c G/Q a Schubert variety. Let L be an ample line bundle on G/Q. Then HI(X, L) = 0 for i > 0 and the restriction map H0(G/Q, L) -+ H0(X, L) is surjective. If char k > 0 this is a consequence of the compatible Frobenius splitting of X in G/Q and the char k = 0 case is handled by semicontinuity. The above result for special Schubert varieties has been proved by SeshadriMusili-Lakshmibai [12], [13] and by Kempf [9] by using characteristic free methods. For X = G/Q, Andersen [1] and Haboush [5] have given simple proofs using characteristic p methods.

On the rank of Hasse-Witt matrix

On the rank of Hasse-Witt matrix

Arithmetic in generalized Artin-Schreier extensions of k( x)

If k is a perfect field of characteristic p ≠ 0 and k( x) is the rational function field over k, it is possible to construct cyclic extensions K n over k( x) such that [ K : k( x)] = p n using the concept of Witt vectors. This is accomplished in the following way; if [ β 1, β 2,…, β n ] is a Witt vector over k( x) = K 0, then the Witt equation y p • y = β generates a tower of extensions through K i = K i−1( y i) where y = [ y 1, y 2,…, y n] . In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in K n . This alternate generation has the form K i = K i−1 ( y i ); y i p − y i = B i , where, as a divisor in K i−1 , B i has the form (B i) = q Πp j λ j . In this form q is prime to Πp j λ j and each λ j is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field K n over an algebraically closed field of constants.

On the Jacobian varieties of hyperelliptic curves over fields of characteristic p > 2

It is well known that an Abelian variety X of dimension g defined over a field K of characteristic p > 0 yields a p-divisible group X(p) of dimension g and of height 2g. Let I’ be the formal group obtained by expansion into power series of the group law of X relative to some system of local parameters at the origin. Then I’ is nothing but the connected p-divisible group in X(p) and r has any height between g and 2g (cf. Tate [14]). In the present paper, we confine ourselves to the study of the Jacobian variety J(C) of a hyperelliptic curve C over a field of characteristic p > 2. Our aims here are (i) to determine the structures of the p-divisible group J(p) and of the formal group r of J(C) (up to isogeny) with the help of the Cartier-Manin matrix A of C, and (ii) to investigate how much information about the algebraic (global) structure of J(C) (up to isogeny) can be recovered from the formal (local) structure. We shall give a brief survey of the paper here. In Section 2, we define the Cartier-Manin matrix A of a hyperelliptic curve C over a perfect field of characteristic p > 2 following Cartier [I] and Manin [S]. We then show that A coincides with the Hasse-Witt matrix of C. Some basic but important properties of A are also discussed. After this, throughout the forthcoming sections, we assume that k is a finite field with pa (a 3 1) elements. In Section 3, we give a complete characterization of the “ordinary” Jacobian variety J(C) of C. When j(C) is ordinary, the Cartier-Manin matrix A of C completely determines the formal structure J(p), and in certain cases the algebraic structure as well (up to isogeny). In the rest of the paper, we study the Jacobian variety J(C) of the hyperelliptic curve C whose Cartier-Manin matrix has determinant zero in R. In Section 4, we observe that the Cartier-Manin matrix A of C no longer provides enough information; it is the p-adic exponents of the eigenvalues of the characteristic polynomial of the Frobenius endomorphism of J(C) relative to k that determine

Residue Free Differentials and the Cartier Operator for Algebraic Function Fields of one Variable

Let K be a field of characteristic p > 0 and let A be a separably generated algebraic function field of one variable with K as its exact constant field. Throughout this paper we shall use the following notations to classify differentials of A/K:D(A) : the K-module of all differentials,G(A) : the K-module of all differentials of the first kind,R(A) : the K-module of all residue free differentials in the sense of Chevalley [2, p. 48],E*(A) : the K-module of all pseudo-exact differentials in the sense of Lamprecht [7, p. 363], (compare the definition with our Lemma 8).

Curves with invertible Hasse-Witt-matrix

Curves with invertible Hasse-Witt-matrix